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Find the equation of parabola whose directrix is \( 0 \) and focus at \( (6, 0) \)
Given, Directrix, focus
Find the equation of parabola whose focus at \( (1, 2) \), directrix \( x2y+3=0\)
Given, , directrix
Equation of parabola is
Find the equation of the set of all points whose distance from \( ( 0, 4) \) are \( \dfrac{2}{3} \) of their distance from the line \( y=9 \).
Let the point be
Distance from
So, the distance from the line is
(Ellipse )
Find the equation of the hyperbola with vertices \( (\pm 5 , 0) \) , foci at \( (\pm 7 , 0) \)
Given, vertices foci
Hence
Equation of hyperbola is
Find the equation of the hyperbola with vertices \( ( 0, \pm 7) \), \( e= \dfrac{4}{3} \)
Vertices ,
Equation of the hyperbola is
The equation of the ellipse having foci \( (0, 1) , (0, 1) \) and minor axis of length \( 1 \) is_______
Given , foci of the ellipse are
Length of minor axis
Equation of the ellipse is
The equation of the parabola having focus at \( ( 3, 5) \) and directrix is \(x2y+3=0 \) is ________
and Directrix :
Let the point on the parabola be
The equation of the hyperbola with vertices \( ( 0, \pm 6) \) and eccentricity \(\dfrac{5}{3} \) is________
Equation of the hyperbola is
The vertices
and
Equation of hyperbola is
The focus of a parabola is \( ( 0, 5) \) and its directrix is \( y=6 \), then equation of parabola is________
Given, and directrix
Let be any point on parabola.
If the parabola \( y^2=4ax \) passes through the point \(( 5, 2) \), the the length of latus rectum is______
Given,
Length of latus rectum
Since the parabola passes through the point
Then ,
If the vertex of the parabola is the point \( (3, 0) \) and the directrix is the line \( x+5=0 \), then its equation is_____
Vertex . Hence and directrix
Since , axis of the parabola is a line perpendicular to directrix and is the midpoint of .
Then,
So,
The equation of the ellipse whose focus is \( (11) \), the directrix is \( xy3 =0 \) and eccentricity is \(\dfrac{1}{2} \) is_____
Focus , Directrix
Let be any point on the ellipse .
The length of the latus rectum of the ellipse is \( 3x^2+y^2=12 \) is________
Given,
Length of latus rectum
If \( e \) is eccentricity of the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 , (a<b) \) then______
We know that,
The eccentricity of the hyperbola whose latus rectum is \( 8 \) and conjugate axis is equal to half of the distance between the foci is_____
Length of latus rectum is (i)
Distance between the foci
(ii)
The common tangents to the circle \( x^2+y^2 =2 \) and the parabola \(y^2 =8x \) touch at the point \( P , Q \) and the parabola at the points \( R ,S \). Then , the area of the quadrilateral \( PQRS \) is______
Let be tangent to
The point of contact being and respectively.
Area of the quadrilateral
sq. units
The ellipse , \( \dfrac{x^2}{9}+\dfrac{y^2}{4}=1 \) is inscribed in a rectangle, whose sides are parallel to the coordinate axes. Another ellipse passing through the point \( (0 ,4 ) \) circumscribes the rectangle. Find the eccentricity of the second ellipse.
Given, , Let
It passes through
It passes through
Also
Let \( (x , y) \) be any point on the parabola \(y^2 =4x \), Let \( P \) be the point that divides the line segment from \( (0, 0) \) to \( (x , y) \) in the ratio \( 1: 3 \). Then the locus of \( P \) is_____
Let , then
Thus the locus of is
Let \( P (6, 3) \) be a point on the hyperbola \( \dfrac{x^2}{a^2}\dfrac{y^2}{b^2}=1 \). If the normal at the point \( P \) intersects the \( x \)axis at \( ( 9, 0)\) then the eccentricity of the hyperbola is______
Equation of normal at is
It passes through , we have
Consider a branch of the hyperbola \( x^22y^22\sqrt{2}x4 \sqrt{2}y6=0 \), with vertex at the point \( A \). Let \( B \) be one of the end points of its latus rectum. If \( C \) is the focus of the hyperbola nearest to the point \( A \), then the area of the triangle \( ABC \) is_____
Given,
We have,
Area of
A hyperbola , having the transverse axis of length \( 2 \sin \theta \), is confocal with the ellipse \( 3x^2+4y^2=12 \). Then its equation is _______
Given,
Required of hyperbola is
The axis of a parabola is along the line \( y=x \) and the distances of its vertex and focus from origin are \(\sqrt{2} \) and \( 2 \sqrt{2} \) respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is______
Vertex, , directrix
Equation of the parabola is
If the line \( 2x+\sqrt{6}y=2 \) touches the hyperbola \( x^22y^2=4 \), then the point of contact is______
Equation of tangent is on comparing with , we get and
The equation of the directrix of the parabola \( y^2+4y +4x+2=0 \) is ______
Given,
Equation of parabola after shifting the origin , where
Equation of directrix is
Required directrix is
The curve described parametrically by \( x=t^2+ t+1, \quad y=t^2 t+1 \), represents_____
We have,
and
Hence the curve represents a parabola.